Real-time optimization of nuclear magnetic resonance experiments

Nuclear Magnetic Resonance (NMR) experiments are typically performed with predetermined pulse sequences and acquisition parameters, and are oftentimes sub-optimal for individual samples under investigation. Here we explore a class of real-time optimization methods that conducts stochastic analyses on the acquired data and in turn updates and optimizes the subsequent measurements. We show superiority of the method to static approaches, both in the efficiency and quality of data acquisition, for a wide range of experiments.

A rapid measurement of T1/T2: the DECPMG sequence

The Driven-Equilibrium Carr-Purcell Meiboom-Gill (DECPMG) pulse sequence is a rapid method for obtaining the average ratio of longitudinal to transverse relaxation times (T(1)/T(2)) as a function of T(2). Since this is a one-dimensional experiment, the (T(1)/T(2))T(2) ratio can be acquired, potentially, in just two scans; the second scan being a reference CPMG measurement. Conventionally, T(1)/T(2) is determined from a two-dimensional T(1)-T(2) relaxation correlation experiment. The method described here offers a significant reduction in experimental time without a reduction in signal-to-noise. The (T(1)/T(2)) ratio is useful for comparing the behaviour of liquids in porous media. Here we demonstrate the application of the DECPMG sequence to the study of oil-bearing rocks by differentiating oil or water saturated rock cores, and by observing the relative strengths of surface interaction for water in two types of rock by measuring (T(1)/T(2)) as a function of magnetic field strength.

The equivalence between off-resonance and on-resonance pulse sequences and its application to steady-state free precession with diffusion in inhomogeneous fields

We show that the spin dynamics of any pulse sequence with off-resonant pulses is identical to that of a modified sequence with on-resonant pulses, including relaxation and diffusion effects. This equivalence applies to pulse sequences with arbitrary offset frequency deltaomega(0) which may exceed the RF field strength omega(1). Using this approach, we examine steady-state free precession (SSFP) in grossly inhomogeneous fields. We show explicitly that the magnitude of the magnetization for each mode at an offset frequency deltaomega(0) is equal to that for SSFP with on-resonance pulses of rescaled amplitude, with the same dependence on relaxation times and diffusion coefficient. The rescaling depends on offset frequency and RF field strength. The theoretical results have been tested experimentally and excellent agreement is found.

Diffusion-relaxation distribution functions of sedimentary rocks in different saturation states

We present diffusion-relaxation distribution functions measured on four rock cores that were prepared in a succession of different saturation states of brine and crude oil. The measurements were performed in a static gradient field at a Larmor frequency of 1.76 MHz. The diffusion-relaxation distribution functions clearly separate the contributions from the two fluid phases. The results can be used to identify the wetting and non-wetting phase, to infer fluid properties of the phases, and to obtain additional information on the geometrical arrangement of the phases. We also observe effects due to restricted diffusion and susceptibility induced internal gradients.

A method for rapid characterization of diffusion

This paper describes a method to determine molecular displacements as a function of time in just two scans: one reference scan using the Carr-Purcell-Meiboom-Gill (CPMG) sequence, a second scan using a modified CPMG sequence (KCPMG). Measurements on free diffusion in bulk fluids, and on restricted diffusion in porous rock samples are reported. This technique can also be used for rapid measurement of flow and chemical exchange.

Quantitative measurement of two-dimensional distribution functions of diffusion and relaxation in grossly inhomogeneous fields

We demonstrate the quantitative extraction of multidimensional distribution functions in the presence of grossly inhomogeneous fields. Examples are shown for diffusion-T(2) distribution functions and T(1)-T(2) distribution functions. The pulse sequences consist of an initial editing sequence followed by a long series of nominal 180 degrees pulses. They are designed such that the kernels describing the relationships between the distribution functions and the measured data are separable. The required phase cycling is discussed. We analyze in detail the extra spin dynamics effects due to the strong field inhomogeneities including the effects on diffusion and relaxation. A recently developed algorithm is used to invert the data and extract stable multidimensional distribution functions in an efficient manner. We present examples for several applications of this new technique. Diffusion-relaxation distribution functions can be used for fluid identification and for the characterization of pore geometry of porous media based on the effects of restricted diffusion. We have also determined T(1)-T(2) distribution functions of water saturated sedimentary rock and find excellent agreement with previous measurements performed in homogeneous fields.

The narrow pulse approximation and long length scale determination in xenon gas diffusion NMR studies of model porous media

We report a systematic study of xenon gas diffusion NMR in simple model porous media, random packs of mono-sized glass beads, and focus on three specific areas peculiar to gas-phase diffusion. These topics are: (i) diffusion of spins on the order of the pore dimensions during the application of the diffusion encoding gradient pulses in a PGSE experiment (breakdown of the narrow pulse approximation and imperfect background gradient cancellation), (ii) the ability to derive long length scale structural information, and (iii) effects of finite sample size. We find that the time-dependent diffusion coefficient, D(t), of the imbibed xenon gas at short diffusion times in small beads is significantly affected by the gas pressure. In particular, as expected, we find smaller deviations between measured D(t) and theoretical predictions as the gas pressure is increased, resulting from reduced diffusion during the application of the gradient pulse. The deviations are then completely removed when water D(t) is observed in the same samples. The use of gas also allows us to probe D(t) over a wide range of length scales and observe the long time asymptotic limit which is proportional to the inverse tortuosity of the sample, as well as the diffusion distance where this limit takes effect (approximately 1-1.5 bead diameters). The Padé approximation can be used as a reference for expected xenon D(t) data between the short and the long time limits, allowing us to explore deviations from the expected behavior at intermediate times as a result of finite sample size effects. Finally, the application of the Padé interpolation between the long and the short time asymptotic limits yields a fitted length scale (the Padé length), which is found to be approximately 0.13b for all bead packs, where b is the bead diameter.

T(1)--T(2) correlation spectra obtained using a fast two-dimensional Laplace inversion

Spin relaxation is a sensitive probe of molecular structure and dynamics. Correlation of relaxation time constants, such as T(1) and T(2), conceptually similar to the conventional multidimensional spectroscopy, have been difficult to determine primarily due to the absense of an efficient multidimensional Laplace inversion program. We demonstrate the use of a novel computer algorithm for fast two-dimensional inverse Laplace transformation to obtain T(1)--T(2) correlation functions. The algorithm efficiently performs a least-squares fit on two-dimensional data with a nonnegativity constraint. We use a regularization method to find a balance between the residual fitting errors and the known noise amplitude, thus producing a result that is found to be stable in the presence of noise. This algorithm can be extended to include functional forms other than exponential kernels. We demonstrate the performance of the algorithm at different signal-to-noise ratios and with different T(1)--T(2) spectral characteristics using several brine-saturated rock samples.

Carr-Purcell sequences with composite pulses

We present novel Carr-Purcell-like sequences using composite pulses that exhibit improved performance in strongly inhomogeneous fields. The sequences are designed to retain the intrinsic error correction of the standard Carr-Purcell-Meiboom-Gill (CPMG) sequence. This is achieved by matching the excitation pulse with the refocusing cycle such that the initial transverse magnetization lies along the axis n(Beta) characterizing the overall rotation of the refocusing cycle. Such sequences are suitable for relaxation measurements. It is shown that in sufficiently inhomogeneous fields, the echo amplitudes have an initial transient modulation that is limited to the first few echoes and then decay with the intrinsic relaxation time of the sample. We show different examples of such sequences that are constructed from simple composite pulses. Sequences of the form 90 degrees (0)-(90 degrees (90-theta/2)-theta(180-theta/2)-90 degrees (90-theta/2))(n) with theta approximately 90 degrees and 270 degrees generate signal over a bandwidth larger than that of the conventional CPMG sequence, resulting in an improved signal-to-noise ratio in inhomogeneous fields. The new sequence 127 degrees (x,y)-(127 degrees (x)-127 degrees (-x))(n) only excites signal off-resonance with a spectrum that is bimodal, peaking at Delta omega(0)=+/-omega(1). Depending on the phase and exact timing of the first pulse, symmetric or antisymmetric excitation is obtained. We also demonstrate several new sequences with improved dependence on the RF field strength. The sequence (22.5 degrees (67.5)-90 degrees (-22.5))-(90 degrees (67.5)-45 degrees (157.5)-90 degrees (67.5))(n) has the property that the phase of the signal depends on B(1), allowing coarse B(1) imaging in a one-dimensional experiment.

Tortuosity measurement and the effects of finite pulse widths on xenon gas diffusion NMR studies of porous media

We have extended the utility of NMR as a technique to probe porous media structure over length scales of approximately 100-2000 microm by using the spin 1/2 noble gas 129Xe imbibed into the system's pore space. Such length scales are much greater than can be probed with NMR diffusion studies of water-saturated porous media. We utilized Pulsed Gradient Spin Echo NMR measurements of the time-dependent diffusion coefficient, D(t), of the xenon gas filling the pore space to study further the measurements of both the pore surface-area-to-volume ratio, S/V(p), and the tortuosity (pore connectivity) of the medium. In uniform-size glass bead packs, we observed D(t) decreasing with increasing t, reaching an observed asymptote of approximately 0.62-0.65D(0), that could be measured over diffusion distances extending over multiple bead diameters. Measurements of D(t)/D(0) at differing gas pressures showed this tortuosity limit was not affected by changing the characteristic diffusion length of the spins during the diffusion encoding gradient pulse. This was not the case at the short time limit, where D(t)/D(0) was noticeably affected by the gas pressure in the sample. Increasing the gas pressure, and hence reducing D(0) and the diffusion during the gradient pulse served to reduce the previously observed deviation of D(t)/D(0) from the S/V(p) relation. The Pade approximation is used to interpolate between the long and short time limits in D(t). While the short time D(t) points lay above the interpolation line in the case of small beads, due to diffusion during the gradient pulse on the order of the pore size, it was also noted that the experimental D(t) data fell below the Pade line in the case of large beads, most likely due to finite size effects.

Optimization of timing in the Carr-Purcell-Meiboom-Gill sequence

The Carr-Purcell-Meiboom-Gill sequence is widely used in grossly inhomogeneous fields to characterize fluid saturated porous media. The distribution of T(2) relaxation times is a measure of the distribution of pore sizes. We present here a theoretical analysis of the spin dynamics of this sequence in strongly inhomogeneous fields. Based on this analysis, we optimize the timing of the sequence for maximal signal bandwidth. It is shown that the initial pulse spacing should be decreased by a time 2t(90)/pi. Experimental results in a strayfield set-up confirm the theoretical analysis. The optimized timing increases the measured signal bandwidth and increases the ratio of signal-to-noise by 1.2 dB without affecting the measured relaxation time.

Diffusion and relaxation effects in general stray field NMR experiments

We analyze the evolution of magnetization following any series of radiofrequency pulses in strongly inhomogeneous fields, with particular attention to diffusion and relaxation effects. When the inhomogeneity of the static magnetic field approaches or exceeds the strength of the RF field, the magnetization has contributions from different coherence pathways. The diffusion or relaxation induced decay of the signal amplitude is in general nonexponential, even if the sample has single relaxation times T(1), T(2) and a single diffusion coefficient D. In addition, the shape of the echo depends on diffusion and relaxation. It is possible to separate contributions from different coherence pathways by phase cycling of the RF pulses. The general analysis is tested on stray field measurements using two different pulse sequences. We find excellent agreement between measurements and calculations. The inversion recovery sequence is used to study the relaxation effects. We demonstrate two different approaches of data analysis to extract the relaxation time T(1). Finite pulse width effects on the timing of the echo formation are also studied. Diffusion effects are analyzed using the Carr--Purcell--Meiboom--Gill sequence. In a stray field of a constant gradient g, we find that unrestricted diffusion leads to nonexponential signal decay versus echo number N, but within experimental error the diffusion attenuation is still only a function of g(2)Dt(3)(E)N, where t(E) is the echo spacing.

Spin dynamics of Carr-Purcell-Meiboom-Gill-like sequences in grossly inhomogeneous B(0) and B(1) fields and application to NMR well logging

The spin dynamics for Carr-Purcell-Meiboom-Gill-like sequences is analyzed in grossly inhomogeneous B(0) and B(1) fields. This problem is important for many applications, especially when the bandwidth of the signal is excitation limited. Examples include stray-field NMR or inside-out NMR probes used in well logging. The amplitudes of the first few echoes exhibit a characteristic transient behavior but quickly approach a smooth asymptotic behavior. For simple Hamiltonians without scalar or dipolar couplings, the evolution of a refocusing subcycle for a given isochromat is described by a rotation. Simple expressions for the signal of the Nth echo are derived in terms of these effective rotations that have a simple geometrical interpretation. It is shown that the asymptotic behavior is controlled by the direction of the axis of these effective rotations and the signal is dominated by magnetization "spin-locked" to the rotation axis. The phase of the signal is independent of the details of the field inhomogeneities. Relaxation in inhomogeneous fields leads to a signal decay that is in general nonexponential with an initial decay rate that is a weighted sum of T(-1)(1) and T(-1)(2). At long times, the echo amplitudes decay to a finite value. Phase cycling eliminates this offset. The effect of diffusion is also studied. This analysis has been applied to an inside-out NMR well logging apparatus. Good quantitative agreement is found between measurements and calculations that are based on the measured B(0) and B(1) field maps.

Dephasing of Hahn echo in rocks by diffusion in susceptibility-induced field inhomogeneities

The decay of the Hahn spin echo of water in the pore space of many porous media is dominated by the dephasing of spins in internal-field inhomogeneities, produced by susceptibility contrasts, rather than surface or bulk relaxation. This is particularly the case for measurements at moderate and high fields in samples such as fluid-saturated sedimentary rocks and some biological materials. Here, we study the behavior of the Hahn-echo decay in rocks with grains much larger and smaller than the average dephasing length, which is typically of the order of a few microns. It is shown that the decay in these two cases is qualitatively different. For coarse-grained rocks, the decay can be modeled to first order by a distribution of local, effective field gradients. This is in contrast to the case of fine-grained rocks, where motional narrowing of the field inhomogeneities occurs. These interpretations are supported by measurements of the temperature dependence of the Hahn echo decay and the diffusion time dependence of the diffusion coefficient.

Effective Gradients in Porous Media Due to Susceptibility Differences

In porous media, magnetic susceptibility differences between the solid phase and the fluid filling the pore space lead to field inhomogeneities inside the pore space. In many cases, diffusion of the spins in the fluid phase through these internal inhomogeneities controls the transverse decay rate of the NMR signal. In disordered porous media such as sedimentary rocks, a detailed evaluation of this process is in practice not possible because the field inhomogeneities depend not only on the susceptibility difference but also on the details of the pore geometry. In this report, the major features of diffusion in internal gradients are analyzed with the concept of effective gradients. Effective gradients are related to the field inhomogeneities over the dephasing length, the typical length over which the spins diffuse before they dephase. For the CPMG sequence, the dependence of relaxation rate on echo spacing can be described to first order by a distribution of effective gradients. It is argued that for a given susceptibility difference, there is a maximum value for these effective gradients, gmax, that depends on only the diffusion coefficient, the Larmor frequency, and the susceptibility difference. This analysis is applied to the case of water-saturated sedimentary rocks. From a set of NMR measurements and a compilation of a large number of susceptibility measurements, we conclude that the effective gradients in carbonates are typically smaller than gradients of current NMR well logging tools, whereas in many sandstones, internal gradients can be comparable to or larger than tool gradients. Copyright 1998 Academic Press.

Diffusion measurement in sandstone core: NMR determination of surface-to-volume ratio and surface relaxivity

Recent theoretical work has shown that the time dependence of the effective diffusion constant in porous media contains information about the surface to pore volume ratio, the tortuosity and other petrophysical information. We have measured the effective diffusion constant in a water saturated core of Fontainebleau sandstone as a function of diffusion time. From the short time behavior, we are able to extract the surface to pore volume ratio. This value can be combined with the spin-lattice relaxation rate measured on the same core under identical conditions to yield the surface relaxivity rho = 1.6 x 10(-5)m/s. Measurements at 25.0 degrees C and 40.0 degrees C demonstrate that the effect of surface relaxation on the measured diffusion coefficient is small, as predicted by theory.